Theorem isnumbasgrplem2 44215

Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)

Assertion

Ref	Expression
isnumbasgrplem2	⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 17400	. . 3 ⊢ Base Fn V
2		ssv 3933	. . 3 ⊢ Grp ⊆ V
3		fvelimab 6962	. . 3 ⊢ ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
4	1, 2, 3	mp2an 705	. 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5		harcl 9560	. . . . . 6 ⊢ (har‘𝑆) ∈ On
6		onenon 10037	. . . . . 6 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (har‘𝑆) ∈ dom card
8		xpnum 10039	. . . . 5 ⊢ (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
9	7, 7, 8	mp2an 705	. . . 4 ⊢ ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10		ssun1 4175	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11		simpr 490	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
12	10, 11	sseqtrrid 3952	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13		fvex 6903	. . . . . . . 8 ⊢ (Base‘𝑥) ∈ V
14	13	ssex 5282	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
15	12, 14	syl 18	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
16	7	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17		simp1l 1216	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18	12	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19		simp2 1155	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3910	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21		ssun2 4176	. . . . . . . . . . 11 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
22	21, 11	sseqtrrid 3952	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23	22	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24		simp3 1156	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
25	23, 24	sseldd 3910	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝑥) = (Base‘𝑥)
27		eqid 2760	. . . . . . . . 9 ⊢ (+g‘𝑥) = (+g‘𝑥)
28	26, 27	grpcl 19161	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
29	17, 20, 25, 28	syl3anc 1398	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
30		simp1r 1217	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
31	29, 30	eleqtrd 2862	. . . . . 6 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32		simplll 787	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
33	22	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34		simprl 783	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
35	33, 34	sseldd 3910	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36		simprr 785	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
37	33, 36	sseldd 3910	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
38	12	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39		simplr 781	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ 𝑆)
40	38, 39	sseldd 3910	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
41	26, 27	grplcan 19220	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
42	32, 35, 37, 40, 41	syl13anc 1399	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
43		simplll 787	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑥 ∈ Grp)
44	12	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45		simprr 785	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ 𝑆)
46	44, 45	sseldd 3910	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ (Base‘𝑥))
47		simprl 783	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
48	44, 47	sseldd 3910	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ (Base‘𝑥))
49	22	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50		simplr 781	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (har‘𝑆))
51	49, 50	sseldd 3910	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (Base‘𝑥))
52	26, 27	grprcan 19193	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
53	43, 46, 48, 51, 52	syl13anc 1399	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
54		harndom 9563	. . . . . . 7 ⊢ ¬ (har‘𝑆) ≼ 𝑆
55	54	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
56	15, 16, 16, 31, 42, 53, 55	unxpwdom3 44206	. . . . 5 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼* ((har‘𝑆) × (har‘𝑆)))
57		wdomnumr 10150	. . . . . 6 ⊢ (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
58	9, 57	ax-mp 5	. . . . 5 ⊢ (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
59	56, 58	sylib 221	. . . 4 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60		numdom 10124	. . . 4 ⊢ ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
61	9, 59, 60	sylancr 599	. . 3 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
62	61	rexlimiva 3155	. 2 ⊢ (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
63	4, 62	sylbi 220	1 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896   ∪ cun 4148   class class class wbr 5102   × cxp 5649  dom cdm 5651   “ cima 5654  Oncon0 6365   Fn wfn 6538  ‘cfv 6543  (class class class)co 7424   ≼ cdom 8978  harchar 9557   ≼^* cwdom 9565  cardccrd 10023  Basecbs 17396  +_gcplusg 17437  Grpcgrp 19153

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-1cn 11265  ax-addcl 11267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-oadd 8487  df-omul 8488  df-er 8724  df-map 8856  df-en 8981  df-dom 8982  df-sdom 8983  df-oi 9511  df-har 9558  df-wdom 9566  df-card 10027  df-acn 10030  df-nn 12343  df-slot 17369  df-ndx 17381  df-base 17397  df-0g 17621  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-grp 19156  df-minusg 19157

This theorem is used by:  isnumbasabl  44217  isnumbasgrp  44218